k-connected graphs

A graph $\Gamma $ is said to be k-connected if $\Gamma $ contains more than k vertices and remains connected if any $k-1$ vertices and their corresponding incident edges are removed. A graph is $\Gamma $ is said to be polyhedral if $\Gamma $ is the $1$-skeleton of a convex polyhedron. According to Steinitz’s theorem, a graph is polyhedral if and only if it is $3$-connected and planar.

Given a polyhedral graph, we have a stronger version of the circle packing theorem [Reference Schramm40] (cf. midsphere for canonical polyhedron).

Marked contact graphs

In many situations, it is better to work with a marking on the graph as well as the circle packing. A marking of a graph $\Gamma $ is a choice of the graph isomorphism

$$ \begin{align*}\phi: \mathscr{G}\longrightarrow \Gamma, \end{align*} $$

where $\mathscr {G}$ is the underlying abstract graph of $\Gamma $. We will refer to the pair $(\Gamma , \phi )$ as a marked graph.

Given two marked plane graphs $(\Gamma _1, \phi _1)$ and $(\Gamma _2, \phi _2)$ with the same underlying abstract graph $\mathscr {G}$, we say that they are equivalent if $\phi _2\circ \phi _1^{-1}:\Gamma _1 \longrightarrow \Gamma _2$ is an isomorphism of plane graphs.

Similarly, a circle packing $\mathcal {P}$ is said to be marked if the associated contact graph is marked.

3.1 Limit set and domain of discontinuity of kissing reflection groups

Let $\mathcal {P} = \{C_1,\ldots , C_n\}$ be a marked circle packing and $D_i$ be the associated open disks for $C_i$. Let P be the set consisting of points of tangency for the circle packing $\mathcal {P}$. Let

$$ \begin{align*}\Pi = \widehat{\mathbb{C}}\setminus\left(\bigcup_{i=1}^n D_i \cup P\right). \end{align*} $$

Then $\Pi $ is a fundamental domain of the action of $G_{\mathcal {P}}$ on the domain of discontinuity $\Omega (G_{\mathcal {P}})$. Denote $\Pi = \bigcup _{i=1}^k \Pi _i$ where $\Pi _i$ is a component of $\Pi $. Note that when the limit set of $G_{\mathcal {P}}$ is connected, each component of the domain of discontinuity is simply connected (that is, a conformal disk) and each component $\Pi _i$ is a closed ideal polygon in the corresponding component of the domain of discontinuity bounded by arcs of finitely many circles in the circle packing.

Since $G_{\mathcal {P}}$ is a free product of finitely many copies of $\mathbb {Z}/2\mathbb {Z}$, each element $g \in G_{\mathcal {P}}$ admits a shortest expression $g = g_{i_1} \ldots g_{i_l}$ ($i_r\neq i_{r+1}$, for $r\in \{1,\cdots , l-1\}$) with respect to the standard generating set $S = \{g_1,\ldots , g_n\}$. The integer l is called the length of the group element g, thought of as a word in terms of the generators.

The following lemma follows directly by induction.

We set $\Pi ^1:= \bigcup _{i=1}^n g_i\cdot \Pi $ and $\Pi ^{j+1} = \bigcup _{i=1}^n g_i\cdot (\Pi ^j\setminus \overline {D}_i)$. For consistency, we also set $\Pi ^0:= \Pi $. We call $\Pi ^l$ the tiling of level l. The following lemma justifies this terminology.

Lemma 3.2.

$$ \begin{align*}\Pi^{l} = \bigcup_{\vert g\vert = l} g\cdot \Pi.\end{align*} $$

Since the domain of discontinuity is the union of tiles of all levels, Lemma 3.2 implies that

$$ \begin{align*}\Omega(G_{\mathcal{P}}) = \bigcup_{g\in G_{\mathcal{P}}} g\cdot \Pi = \bigcup_{i=0}^\infty \Pi^i. \end{align*} $$

Similarly, we set $\mathcal {D}^i = \widehat {\mathbb {C}} \setminus \bigcup _{j=0}^i\Pi ^j$. We remark that $\mathcal {D}^i$ is neither open nor closed: it is a finite union of open disks together with the orbit of P under the group elements of length up to i on the boundaries of these disks.

Let $\overline {\mathcal {D}^i}$ be its closure. Note that each $\overline {\mathcal {D}^i}$ is a union of closed disks for some (possibly disconnected) finite circle packing. Indeed,

$$ \begin{align*}\overline{\mathcal{D}^0} = \overline{\widehat{\mathbb{C}}\setminus\Pi^0} = \bigcup_{i=1}^n \overline{D}_i \end{align*} $$

is the union of the closed disks corresponding to the original circle packing $\mathcal {P}$. By induction, we have that at level $i+1$,

(3.1)$$ \begin{align} \overline{\mathcal{D}^{i+1}} = \bigcup_{j=1}^n g_j \cdot \overline{\mathcal{D}^i\setminus\overline{D}_j} \end{align} $$

is the union of the images of the level i disks outside of $\overline {D}_j$ under $g_j$.

We also note that the sequence $\overline {\mathcal {D}}^i$ is nested and thus the limit set

$$ \begin{align*}\Lambda(G_{\mathcal{P}}) = \bigcap_{i=0}^\infty \mathcal{D}^i = \bigcap_{i=0}^\infty \overline{\mathcal{D}^i}. \end{align*} $$

Therefore, we have the following expansive property of the group action on $\Lambda (G_{\mathcal {P}})$.

We now prove the group part of Theorem 1.1.

Proof. If $\Gamma $ is not $2$-connected, then there exists a circle (say $C_1$) such that the circle packing becomes disconnected once we remove it. Then we see $\mathcal {D}^1 \subseteq \widehat {\mathbb {C}}$ is disconnected by Equation 3.1. This forces the limit set to be disconnected as well (see Figure 3.1).

Figure 3.1 A disconnected limit set for a kissing reflection group G with non-$2$-connected contact graph. G is generated by reflections along the five visible large circles in the figure.

On the other hand, if $\Gamma $ is $2$-connected, then $\mathcal {D}^1$ is connected by Equation 3.1. Now by induction and Equation 3.1 again, $\mathcal {D}^i$ is connected for all i. Thus, $\Lambda (G_{\mathcal {P}}) = \bigcap _{i=0}^\infty \mathcal {D}^i$ is also connected.

Proposition 3.4 and the definition of kissing reflection groups show that the association of a $2$-connected simple plane graph with a kissing reflection group with connected limit set is well defined and surjective. To verify that this is indeed injective, we remark that if $\mathcal {P}$ and $\mathcal {P}'$ are two circle packings associated to two contact graphs that are nonisomorphic as plane graphs, then the closures of the fundamental domains $\overline {\Pi }$ and $\overline {\Pi '}$ are not homeomorphic. Note that the touching patterns of different components of $\Pi $ or $\Pi '$ completely determine the structures of the pairing cylinders of the associated $3$-manifolds at the cusps (See [Reference Marden25, §2.6]). This means that the conformal boundaries of $\mathbb {H}^3/\widetilde {G}_{\mathcal {P}}$ and $\mathbb {H}^3/\widetilde {G}_{\mathcal {P}'}$ with the pairing cylinder structures are not the same. Thus, the two kissing reflection groups $G_{\mathcal {P}}$ and $G_{\mathcal {P}'}$ are not quasiconformally isomorphic.

3.2 Acylindrical kissing reflection groups

Recall that $\widetilde {G}_{\mathcal {P}}$ is the index $2$ subgroup of $G_{\mathcal {P}}$ consisting of orientation-preserving elements. We set

$$ \begin{align*}\mathcal{M}(G_{\mathcal{P}}) := \mathbb{H}^3 \cup \Omega(G_{\mathcal{P}})/\widetilde{G}_{\mathcal{P}} \end{align*} $$

to be the associated $3$-manifold with boundary. Note that the boundary

$$ \begin{align*}\partial \mathcal{M}(G_{\mathcal{P}}) = \Omega(G_{\mathcal{P}})/\widetilde{G}_{\mathcal{P}} \end{align*} $$

is a finite union of punctured spheres. Each punctured sphere corresponds to the double of a component of $\Pi $.

Let F be a face of $\Gamma $. Then it corresponds to a component $\Pi _F$ of $\Pi $, which also corresponds to a component $R_F$ of $\partial \mathcal {M}(G_{\mathcal {P}})$. More precisely, there is a unique component $\Omega _F$ of $\Omega (G_{\mathcal {P}})$ containing $\Pi _F$ and

$$ \begin{align*}R_F\cong \Omega_F / \text{stab}(\Omega_F), \end{align*} $$

where $\text {stab}(\Omega _F)$ is the stabiliser of $\Omega _F$ in $\widetilde {G}_{\mathcal {P}}$.

A compact $3$-manifold $M^3$ with boundary is called acylindrical if $M^3$ contains no essential cylinders and is boundary incompressible. Here an essential cylinder C in $M^3$ is a closed cylinder C such that $C\cap \partial M^3 = \partial C$, the boundary components of C are not homotopic to points in $\partial M^3$ and C is not homotopic into $\partial M^3$. $M^3$ is said to be boundary incompressible if the inclusion is injective for every component R of $\partial M^3$. (We refer the reader to [Reference Marden26, §3.7, §4.7] for detailed discussions.)

Our manifold $\mathcal {M}(G_{\mathcal {P}})$ is not a compact manifold as there are parabolic elements (cusps) in $\widetilde {G}_{\mathcal {P}}$. Thurston [Reference Thurston45] introduced the notion of pared manifolds to work with Kleinian groups with parabolic elements. In our setting, we can also use an equivalent definition without introducing pared manifolds. To start the definition, we note that for a geometrically finite group, associated with the conjugacy class of a rank 1 cusp, there is a pair of punctures $p_1, p_2$ on $\partial M^3$. If $c_1, c_2$ are small circles in $\partial M$ retractable to $p_1, p_2$, then there is a pairing cylinder C in $M^3$, which is a cylinder bounded by $c_1$ and $c_2$ (see [Reference Marden25, §2.6], [Reference Marden26, p. 125]).

Note that $\mathcal {M}(G_{\mathcal {P}})$ is boundary incompressible if and only if each component of $\Omega (G_{\mathcal {P}})$ is simply connected if and only if the limit set $\Lambda (G_{\mathcal {P}})$ is connected. We also note that the acylindrical condition is a quasiconformal invariant and hence does not depend on the choice of the circle packing $\mathcal {P}$ realising a simple connected plane graph $\Gamma $. In the remainder of this section, we shall prove the following characterisation of acylindrical kissing Kleinian reflection groups.

This proposition will be proved after the following lemmas. Let $\Gamma $ be a $2$-connected simple plane graph and $\mathcal {P} = \{C_1,\ldots , C_n\}$ be a realisation of $\Gamma $. Let $G_{\mathcal {P}}$ be the kissing reflection group, with generators $g_1,\ldots , g_n$ given by reflections along $C_1,\ldots , C_n$. Note that a face F of $\Gamma $ corresponds to a component $R_F$ of $\partial \mathcal {M}(G_{\mathcal {P}})$.

Any two nonadjacent vertices $v, w$ of the face F give rise to an essential simple closed curve $\widetilde {\gamma }^F_{vw}$ on $R_F$ (a simple closed curve on a surface is essential if it is not homotopic to a point or a puncture). More precisely, let $g_v, g_w$ be the reflections associated to the two vertices; then $g_vg_w \in \text {stab}(\Omega _F)$ is a loxodromic element under the uniformisation of $R_F$ which gives the simple closed curve $\widetilde {\gamma }^F_{vw}$ on $R_F$. Note that $g_vg_w$ itself may not be loxodromic as the vertices $v,w$ may be adjacent in some other faces. If this is the case, then we have an accidental parabolic element (see [Reference Marden26, p. 198, Section 3, Problem 17]).

We first prove the following graph-theoretic lemma.

Proof. As $\Gamma $ is not $3$-connected, there exist two vertices $v, w$ so that $\Gamma \setminus \{v, w\}$ is disconnected. Let $F_1,\ldots , F_k$ be the list of faces that contain v as a vertex. Since $\Gamma $ is plane, we may assume that the faces $F_i$ are ordered around v counterclockwise. Since $\Gamma $ is plane and $2$-connected, each face $F_i$ is a Jordan domain. Let $v_{i,0} = v, v_{i,1},\ldots , v_{i, j_i}$ be the vertices of $F_i$ ordered counterclockwise. Since the faces $F_i$ are ordered counterclockwise, we have that $v_{i,j_i} = v_{i+1, 1}$. We remark that there might be additional identifications. Then

$$ \begin{align*}v_{1,1}\to v_{1,2}\to \ldots \to v_{1,j_1} (=v_{2,1}) \to v_{2,2}\to \ldots \to v_{k,j_k} = v_{1,1} \end{align*} $$

form a (potentially non-simple) cycle C (see Figure 3.2). Since $\Gamma $ is $2$-connected, $\Gamma \setminus \{w\}$ is connected. Thus, in particular, any vertex p is connected to $C\setminus \{w\}$ in $\Gamma \setminus \{w\}$. Thus, if $w \notin C$ or w only appears once in C, then $C\setminus \{w\}$ is connected. This would imply that $\Gamma \setminus \{v,w\}$ is connected, which is a contradiction. Therefore, w must appear at least twice in the cycle C. Since each face is a Jordan domain, w must appear on the boundaries of at least two faces $F_{i_1}$ and $F_{i_2}$.

Figure 3.2 A schematic of the potentially non-simple cycle around v.

Since $\Gamma $ is simple, w is adjacent to v in at most two faces, in which case $w = v_{i,j_i} = v_{i+1,1}$; that is, it contributes to only one point in C. Therefore, there exists a face on which w is not adjacent to v. This proves the lemma.

We can now prove one direction of Proposition 3.6.

Proof. Note that $\Gamma $ must be $2$-connected as $\Lambda (G_{\mathcal {P}})$ is connected (by the boundary incompressibility condition). We will prove the contrapositive and assume that $\Gamma $ is not $3$-connected. Let $v,w$ be the two vertices given by Lemma 3.7. There are two cases.

If $v, w$ are nonadjacent vertices in two faces $F_1, F_2$, then $g_vg_w$ gives a pair of essential simple closed curves on $R_{F_1}$ and $R_{F_2}$ in $\partial \mathcal {M}(G_{\mathcal {P}})$. This pair bounds an essential cylinder (see the Cylinder Theorem in [Reference Marden26, §3.7]), which is not homotopic to a pairing cylinder of two punctures (see $g_Cg_{C''}$ in Figure 3.4).

Figure 3.3 The limit set of a kissing reflection group G with a $3$-connected contact graph.

Figure 3.4 The limit set of a kissing reflection group G with Hamiltonian but non-$3$-connected contact graph. The unique Hamiltonian cycle of the associated contact graph $\Gamma $ divides the fundamental domain $\Pi (G)$ into two parts $\Pi ^\pm $, which are shaded in grey and blue. With appropriate markings, G is the mating of two copies of the group H shown in Figure 3.6.

If $v, w$ are nonadjacent vertices in $F_1$ but adjacent vertices in $F_2$, then $g_vg_w$ corresponds to an essential simple closed curve in $R_{F_1}$ and a simple closed curve homotopic to a puncture in $R_{F_2}$. Then $g_vg_w$ is an accidental parabolic and the two curves bound an essential cylinder which is not homotopic to a pairing cylinder (see $g_Cg_{C'}$ in Figure 3.4).

Therefore, in either case, $G_{\mathcal {P}}$ is cylindrical.

Gasket limit set

Recall that a closed set $\Lambda $ is a round gasket if

• • $\Lambda $ is the closure of some infinite circle packing; and

• • the complement of $\Lambda $ is a union of round disks which is dense in $\widehat {\mathbb {C}}$.

We call a homeomorphic copy of a round gasket a gasket. See Section 2 for our definition of infinite circle packings.

If $\Gamma $ is $3$-connected, then $\Gamma $ is a polyhedral graph. Let $\Gamma ^{\vee }$ be the planar dual of $\Gamma $. Then Theorem 2.2 gives a (unique) pair of circle packings $\mathcal {P}$ and $\mathcal {P}^{\vee }$ whose contact graphs are isomorphic to $\Gamma $ and $\Gamma ^{\vee }$ (respectively) as plane graphs such that $\mathcal {P}$ and $\mathcal {P}^{\vee }$ intersect orthogonally at their points of tangency (see Figure 3.3). Let $G_{\mathcal {P}}$ be the kissing reflection group associated with $\mathcal {P}$. Since the circle packing $\mathcal {P}^{\vee }$ is dual to $\mathcal {P}$, we have that

$$ \begin{align*}\bigcup_{g\in G_{\mathcal{P}}} \bigcup_{C\in \mathcal{P}^{\vee}} g\cdot C \end{align*} $$

is an infinite circle packing and the limit set is the closure

$$ \begin{align*}\Lambda(G_{\mathcal{P}}) = \overline{\bigcup_{g\in G_{\mathcal{P}}} \bigcup_{C\in \mathcal{P}^{\vee}} g\cdot C}. \end{align*} $$

Since $\Lambda (G_{\mathcal {P}})$ is nowhere dense and the complement is a union of round disks, we conclude that $\Lambda (G_{\mathcal {P}})$ is a round gasket.

Note that each component of $\Omega (G_{\mathcal {P}})$ is of the form $g\cdot D$ where $g\in G_{\mathcal {P}}$ and D is a disk in the dual circle packing $\mathcal {P}^{\vee }$. By induction, we have the following.

We have the following characterisation of gasket limit set for kissing reflection groups.

In the course of the proof, we have actually derived the following characterisation which is worth mentioning.

We are now able to prove the other direction of Proposition 3.6.

We remark that the unique configuration of pairs of circle packings given in Theorem 2.2 gives a kissing reflection group with totally geodesic boundary. This unique point in the deformation space of acylindrical manifolds is guaranteed by a theorem of McMullen [Reference McMullen28].

3.3 Deformation spaces of kissing reflection groups

Throughout this section, we will use bold symbols, such as $\mathbf {G}, \mathbf {G}_\Gamma $, to represent the base point for the corresponding deformation spaces. We use regular symbols G to represent the image of a representation in the deformation spaces. If the group is a kissing reflection group, we also use $G_{\mathcal {P}}$ if we want to emphasise the corresponding circle packing $\mathcal {P}$.

Definition of $\textrm {AH}(\mathbf {G})$

Let $\mathbf {G}$ be a finitely generated discrete subgroup of $\textrm {Aut}^\pm (\widehat {\mathbb {C}})$. A representation (that is, a group homomorphism) $\xi : \mathbf {G} \longrightarrow \textrm {Aut}^\pm (\widehat {\mathbb {C}})$ is said to be weakly type-preserving

1. 1. if $\xi (g) \in \textrm {Aut}^+(\widehat {\mathbb {C}})$ if and only if $g\in \textrm {Aut}^+(\widehat {\mathbb {C}})$ and

2. 2. if $g \in \textrm {Aut}^+(\widehat {\mathbb {C}})$, then $\xi (g)$ is parabolic whenever g is parabolic.

Note that a weakly type-preserving representation may send a loxodromic element to a parabolic one.

Definition 3.13. Let $\mathbf {G}$ be a finitely generated discrete subgroup of $\textrm {Aut}^\pm (\widehat {\mathbb {C}})$.

$$ \begin{align*} \textrm{AH}(\mathbf{G}) &:= \lbrace\xi: \mathbf{G}\longrightarrow G\ \textrm{is a weakly type-preserving isomorphism to} \\ & \textrm{a discrete subgroup}\ G\ \textrm{of}\ \textrm{Aut}^\pm(\widehat{\mathbb{C}})\rbrace / \sim,\; \end{align*} $$

where $\xi _1\sim \xi _2$ if there exists a Möbius transformation M such that

$$ \begin{align*}\xi_2(g)=M\circ\xi_1(g)\circ M^{-1},\ \textrm{for all}\ g\in \mathbf{G}.\end{align*} $$

$\textrm {AH}(\mathbf {G})$ inherits the quotient topology of algebraic convergence. Indeed, we say that a sequence of weakly type-preserving representations $\{\xi _n\}$ converges to $\xi $ algebraically if $\{\xi _n(g_i)\}$ converges to $\xi (g_i)$ as elements of $\textrm {Aut}^\pm (\widehat {\mathbb {C}})$ for (any) finite generating set $\{g_i\}$.

Quasiconformal deformation space of $\mathbf {G}$

Recall that a Kleinian group is said to be geometrically finite if it has a finite-sided fundamental polyhedron. We say a finitely generated discrete subgroup of $\textrm {Aut}^\pm (\widehat {\mathbb {C}})$ is geometrically finite if the index $2$ subgroup is geometrically finite.

Let $\mathbf {G}$ be a finitely generated, geometrically finite, discrete subgroup of $\textrm {Aut}^\pm (\widehat {\mathbb {C}})$. A group G is called a quasiconformal deformation of $\mathbf {G}$ if there is a quasiconformal map $F: \widehat {\mathbb {C}} \longrightarrow \widehat {\mathbb {C}}$ that induces an isomorphism $\xi : \mathbf {G} \longrightarrow G$ such that $F \circ g(z) = \xi (g) \circ F(z)$ for all $g\in \mathbf {G}$ and $z\in \widehat {\mathbb {C}}$. Such a group G is necessarily geometrically finite and discrete. The map F is uniquely determined (up to normalisation) by its Beltrami differential on the domain of discontinuity $\Omega (\mathbf {G})$ (See [Reference Marden26, §5.1.2, Theorem 3.13.5]).

We define the quasiconformal deformation space of $\mathbf {G}$ as

$$ \begin{align*}\mathcal{QC}(\mathbf{G}) = \{\xi \in \textrm{AH}(\mathbf{G}): \xi \text{ is induced by a quasiconformal deformation of } \mathbf{G}\}. \end{align*} $$

By definition, $\mathcal {QC}(\mathbf {G}) \subseteq \textrm {AH}(\mathbf {G})$.

Kissing reflection groups

The above discussion is quite general and applies to any Kleinian (reflection) group. In the following, we will specialise to the case of a kissing reflection group.

Recall that different realisations of a fixed marked, connected simple plane graph $\Gamma $ as circle packings $\mathcal {P}$ give canonically isomorphic kissing reflection groups $G_{\mathcal {P}}$. Thus, the algebraic/quasiconformal deformation spaces of all such $G_{\mathcal {P}}$ can be canonically identified. However, in order to study these deformation spaces, it will be convenient to fix a (marked) circle packing realisation $\mathcal {P}$ of the (marked) graph $\Gamma $ and use the associated kissing reflection group

$$ \begin{align*}\mathbf{G}_\Gamma:=G_{\mathcal{P}} \end{align*} $$

as the base point. With this choice, we can and will use the notation $\textrm {AH}(\Gamma )$ and $\mathcal {QC}(\Gamma )$ to denote $\textrm {AH}(\mathbf {G}_\Gamma )$ and $\mathcal {QC}(\mathbf {G}_\Gamma )$, respectively.

Now let $\Gamma _0$ be a simple plane graph. The goal of this section is to describe the algebraic deformation space $\textrm {AH}(\Gamma _0)$ and the closure $\overline {\mathcal {QC}(\Gamma _0)}$ (of the quasiconformal deformation space) in $\textrm {AH}(\Gamma _0)$.

In line with the convention introduced above, we choose a circle packing $\{\mathbf {C}_1,\ldots , \mathbf {C}_n\}$ realising $\Gamma _0$ and denote the reflection along $\mathbf {C}_i$ as $\rho _i$. Let $\mathbf {G}_{\Gamma _0}$ be the kissing reflection group generated by the reflections $\rho _i$. We remark that since we will be working with deformation spaces for several contact graphs, we use $\mathbf {G}_{\Gamma _0}$ to emphasise that the contact graph is $\Gamma _0$.

Let G be a discrete subgroup of $\textrm {Aut}^\pm (\widehat {\mathbb {C}})$ and let $\xi : \mathbf {G}_{\Gamma _0}\longrightarrow G$ be a weakly type-preserving isomorphism. Since the union of the circles $\mathbf {C}_1,\ldots , \mathbf {C}_n$ is connected, for each $\rho _i$ there exists $\rho _j$ so that $\rho _i\circ \rho _j$ is parabolic. This implies that no $\xi (\rho _i)$ is the antipodal map. Hence, $\xi (\rho _i)$ is also a circular reflection. Assume that $\xi (\rho _i)$ is the reflection along some circle $C_i$. If $\rho _i \circ \rho _j$ is parabolic, then $\xi (\rho _i) \circ \xi (\rho _{j})$ is also parabolic. Thus, $C_i$ is tangent to $C_j$ as well. This motivates the following definition.

Definition 3.14. Let $\Gamma $ be a simple plane graph with the same number of vertices as $\Gamma _0$. We say that $\Gamma $ abstractly dominates $\Gamma _0$, denoted by $\Gamma \geq _{a} \Gamma _0$, if there exists an embedding $\psi : \Gamma _0 \longrightarrow \Gamma $ as abstract graphs (that is, if there exists a graph isomorphism between $\Gamma _0$ and a subgraph of $\Gamma $).

We say that $\Gamma $ dominates $\Gamma _0$, denoted by $\Gamma \geq \Gamma _0$, if there exists an embedding $\psi : \Gamma _0 \longrightarrow \Gamma $ as plane graphs (that is, if there exists a graph isomorphism between $\Gamma _0$ and a subgraph of $\Gamma $ that extends to an orientation-preserving homeomorphism of $\widehat {\mathbb {C}}$).

We also define

$$ \begin{align*}\mathrm{Emb}_a(\Gamma_0):=\{(\Gamma, \psi): \Gamma\geq_a \Gamma_0 \text{ and } \psi: \Gamma_0 \longrightarrow \Gamma \text{ is an embedding as abstract graphs}\} \end{align*} $$

and

$$ \begin{align*}\mathrm{Emb}_p(\Gamma_0):=\{(\Gamma, \psi): \Gamma\geq \Gamma_0 \text{ and } \psi: \Gamma_0 \longrightarrow \Gamma \text{ is an embedding as plane graphs}\}. \end{align*} $$

In other words, a simple plane graph $\Gamma $ abstractly dominates $\Gamma _0$ if $\Gamma $ (as an abstract graph) can be constructed from $\Gamma _0$ by introducing new edges, and it dominates $\Gamma _0$ if one can do this while respecting the plane structure.

We emphasise that the graph $\Gamma $ is always assumed to be planar, but the embedding $\psi $ may not respect the plane structure in $\mathrm {Emb}_a(\Gamma _0)$ (see Figure 3.5). We also remark that an element $(\Gamma , \psi )$ is identified with $(\Gamma ', \psi ')$ in $\mathrm {Emb}_a(\Gamma _0)$ or $\mathrm {Emb}_p(\Gamma _0)$ if $\psi '\circ \psi ^{-1}$ extends to an isomorphism between $\Gamma $ and $\Gamma '$ as plane graphs.

Figure 3.5 The graph on the left abstractly dominates the graph on the right, but no embedding of the right graph into the left graph respects the plane structure.

We have the following lemma.

Note that for kissing reflection groups, the double of the polyhedron bounded by the half-planes associated to the circles $C_i$ is a fundamental polyhedron for the action of the index two Kleinian group on $\mathbb {H}^3$. Thus, we have the following corollary which is worth mentioning.

Let $(\Gamma , \psi ) \in \mathrm {Emb}_p(\Gamma _0)$ (respectively in $\mathrm {Emb}_a(\Gamma _0)$). Let us fix a circle packing $C_{1,\Gamma }, \ldots , C_{n, \Gamma }$ realising $\Gamma $, where $C_{i,\Gamma }$ corresponds to $\mathbf {C}_i$ under the embedding $\psi $ of $\Gamma _0$. Let $g_i$ be the associated reflection along $C_{i, \Gamma }$ and $\mathbf {G}_\Gamma = \langle g_1,\ldots , g_{n}\rangle $. Then Lemma 3.15 shows that

$$ \begin{align*} \xi_{(\Gamma,\psi)}: \mathbf{G}_{\Gamma_0} &\longrightarrow \mathbf{G}_\Gamma\\ \rho_i &\mapsto g_i \end{align*} $$

is a weakly type-preserving isomorphism. Thus, $\mathcal {QC}(\Gamma )= \mathcal {QC}(\mathbf {G}_\Gamma )$ can be embedded in $\textrm {AH}(\Gamma _0)= \textrm {AH}(\mathbf {G}_{\Gamma _0})$. Indeed, if $\xi : \mathbf {G}_\Gamma \longrightarrow G$ represents an element in $\mathcal {QC}(\Gamma )$, then

$$ \begin{align*}\xi\circ \xi_{(\Gamma,\psi)}: \mathbf{G}_{\Gamma_0} \longrightarrow G \end{align*} $$

is a weakly type-preserving isomorphism. It can be checked that the map $\xi \mapsto \xi \circ \xi _{(\Gamma ,\psi )}$ gives an embedding of $\mathcal {QC}(\Gamma )$ into $\textrm {AH}(\Gamma _0)$. We shall identify the space $\mathcal {QC}(\Gamma )$ with its image in $\textrm {AH}(\Gamma _0)$ under this embedding.

Proposition 3.17.

$$ \begin{align*}\textrm{AH}(\Gamma_0) = \bigcup_{(\Gamma, \psi) \in \mathrm{Emb}_a(\Gamma_0)} \mathcal{QC}(\Gamma) \end{align*} $$

and

$$ \begin{align*}\overline{\mathcal{QC}(\Gamma_0)} = \bigcup_{(\Gamma, \psi) \in \mathrm{Emb}_p(\Gamma_0)} \mathcal{QC}(\Gamma). \end{align*} $$

The proof of this proposition will be furnished after a discussion of pinching deformations. Once the connection with pinching deformation is established, the result can be derived from [Reference Ohshika35].

The perspective of pinching deformation

For the general discussion of pinching deformations, let us fix an arbitrary $2$-connected simple plane graph $\Gamma $, a circle packing $\mathcal {P}$ with contact graph isomorphic to $\Gamma $ and the associated kissing reflection group $\mathbf {G}= G_{\mathcal {P}}$, which we think of as the base point of the quasiconformal deformation space $\mathcal {QC}(\Gamma )$. Since $\Gamma $ is the only graph appearing in this discussion (until the proof Proposition 3.17), we omit the subscript ‘$\Gamma $’ from the base point $\mathbf {G}$.

Let F be a face of $\Gamma $ and $R_F$ be the associated component of $\partial \mathcal {M}(\mathbf {G})$. Let $C\in \mathcal {P}$ be a circle on the boundary of $\Pi _F$. Then the reflection along C descends to an anti-conformal involution on $R_F$, which we shall denote as

$$ \begin{align*}\sigma_F: R_F \longrightarrow R_F. \end{align*} $$

Note that different choices of the circle descend to the same involution. It is known that for boundary incompressible geometrically finite Kleinian groups, the quasiconformal deformation space is the product of the Teichmüller spaces of the components of the conformal boundary (see [Reference Marden26, Theorem 5.1.3]):

$$ \begin{align*}\mathcal{QC}(\widetilde{\mathbf{G}}) = \prod_{F \text{ face of } \Gamma} \textrm{Teich} (R_F), \end{align*} $$

where $\widetilde {\mathbf {G}}$ is the index two subgroup of $\mathbf {G}$ consisting of orientation-preserving elements.

We denote by $\textrm {Teich}^{\sigma _F} (R_F) \subseteq \textrm {Teich} (R_F)$ those elements corresponding to $\sigma _F$-invariant quasiconformal deformations of $R_F$. Then

$$ \begin{align*}\mathcal{QC}(\mathbf{G}) = \prod_{F \text{ face of } \Gamma} \textrm{Teich}^{\sigma_F} (R_F). \end{align*} $$

Indeed, any element in $\prod _{F \text { face of } \Gamma } \textrm {Teich}^{\sigma _F} (R_F)$ is uniquely determined by the associated Beltrami differential on $\Pi _F$, which can be pulled back by $\mathbf {G}$ to produce a $\mathbf {G}$-invariant Beltrami differential on $\widehat {\mathbb {C}}$. Such a Beltrami differential can be uniformised by the measurable Riemann mapping theorem.

Recall that $\Pi _F$ is an ideal polygon. Note that the component of $\Omega (\mathbf {G})$ containing $\Pi _F$ is simply connected and hence $\Pi _F$ inherits the hyperbolic metric from the corresponding component of $\Omega (\mathbf {G})$. Then $R_F$ is simply the double of $\Pi _F$. We claim that the only $\sigma _F$-invariant geodesics are those simple closed curves $\widetilde {\gamma }^F_{vw}$ on $R_F$ (see the discussion following Proposition 3.6 for the definition of $\widetilde {\gamma }^F_{vw}$), where $v,w$ are two nonadjacent vertices on the boundary of F. Indeed, any $\sigma _F$-invariant geodesic would intersect the boundary of the ideal polygon perpendicularly and the $\widetilde {\gamma }^F_{vw}$ are the only geodesics satisfying this property. We denote the associated geodesic arc in $\Pi _F$ by $\gamma ^F_{vw}$.

We define a multicurve on a surface as a disjoint union of simple closed curves, such that no two components are homotopic. It is said to be weighted if a nonnegative number is assigned to each component. We shall identify two multicurves if they are homotopic to each other.

Let $\mathcal {T}$ be a triangulation of F obtained by adding new edges connecting the vertices of F. Since each additional edge in this triangulation connects two nonadjacent vertices of F, we can associate a multicurve $\widetilde {\alpha }^F_{\mathcal {T}}$ on $R_F$ consisting of all $\widetilde {\gamma }^F_{vw}$, where $vw$ is a new edge in $\mathcal {T}$. A marking of the graph $\Gamma $ also gives a marking of the multicurve. We use $\alpha ^F_{\mathcal {T}}$ to denote the multi-arc in the hyperbolic ideal polygon $\Pi _F$. Since $\mathcal {T}$ is a triangulation, the multicurve $\widetilde {\alpha }^F_{\mathcal {T}}$ yields a pants decomposition of the punctured sphere $R_F$ such that the boundary components of every pair of pants are punctures or simple closed curves in $\widetilde {\alpha }^F_{\mathcal {T}}$. Thus, the complex structures of these pairs of pants are uniquely determined by the lengths of the marked multicurve $\widetilde {\alpha }^F_{\mathcal {T}}$ (cf. [Reference Imayoshi and Taniguchi15, Theorem 3.5]). Since $R_F$ is invariant under the anti-conformal involution $\sigma _F$, the surface $R_F$ is uniquely determined by the complex structures of the above pairs of pants and hence by the lengths of the marked multicurve $\widetilde {\alpha }^F_{\mathcal {T}}$. If F has m sides, then any triangulation of F has $m-3$ additional edges. Thus, we have the analogue of Fenchel–Nielsen coordinates on

$$ \begin{align*}\textrm{Teich}^{\sigma_F} (R_F) = \mathbb{R}_+^{m-3} \end{align*} $$

by assigning to each element of $\textrm {Teich}^{\sigma _F} (R_F)$ the lengths of the marked multicurve $\widetilde {\alpha }^F_{\mathcal {T}}$ (or, equivalently, the lengths of the marked multi-arc $\alpha ^F_{\mathcal {T}}$). Note that with these lengths, the multi-arcs and the multicurves become weighted. Note that different triangulations yield different coordinates. We also remark that unlike classical Fenchel–Nielsen coordinates on Teichmüller spaces, Fenchel–Nielsen coordinates on $\textrm {Teich}^{\sigma _F} (R_F)$ do not contain twist parameters precisely due to the $\sigma _F-$invariance of quasiconformal deformations (cf. [Reference Imayoshi and Taniguchi15, §3.2]).

In order to degenerate in $\textrm {Teich}^{\sigma _F} (R_F)$, the length of some arc $\gamma ^F_{vw}$ must shrink to $0$. The above discussion implies that the closure of $\textrm {Teich}^{\sigma _F} (R_F)$ in the Thurston compactification consists only of weighted $\sigma _F$-invariant multicurves $\widetilde {\alpha }^F$. If $S_k \in \textrm {Teich}^{\sigma _F} (R_F)$ converges to a weighted $\sigma _F$-invariant multicurve $\widetilde {\alpha }^F$, then we say that $S_k$ is a pinching deformation on $\widetilde {\alpha }^F$. More generally, we say that $G_k \in \mathcal {QC}(\mathbf {G})$ is a pinching deformation on $\widetilde {\alpha } = \bigcup \widetilde {\alpha }^F$ if for each face F, the conformal boundary associated to F is a pinching deformation on $\widetilde {\alpha }^F$.

Given two curves $\widetilde {\gamma }^F_{vw} \subseteq R_F$ and $\widetilde {\gamma }^{F'}_{ut} \subseteq R_{F'}$, we say that they are parallel if they are homotopic in the $3$-manifold $\mathcal {M}(G_{\mathcal {P}})$. Since the curve $\widetilde {\gamma }^F_{vw}$ corresponds to the element $g_vg_w$, we get that $\widetilde {\gamma }^F_{vw}$ and $\widetilde {\gamma }^{F'}_{ut}$ are parallel if and only if (possibly after switching the order) $v=u$ and $w=t$. Note that in this case, $g_vg_w \in \text {stab}(\Omega _F) \cap \text {stab}(\Omega _{F'})$ and $v, w$ are on the common boundaries of F and $F'$. (See, for example, Figure 3.4 where $v, w$ correspond to the circles $C, C''$, where the curve $\widetilde {\gamma }^{F_1}_{vw}$ is parallel with $\widetilde {\gamma }^{F_2}_{vw}$.)

Let $\widetilde {\alpha } = \bigcup \widetilde {\alpha }^F$ be a union of $\sigma _F$-invariant multicurves. We say that $\widetilde {\alpha }$ is a nonparallel multicurve if no two components are parallel. We note that nonparallel multicurves $\widetilde {\alpha }$ for $\mathbf {G}$ are in one-to-one correspondence with the simple plane graphs that dominate $\Gamma $. Indeed, any multicurve $\widetilde {\alpha } = \bigcup \widetilde {\alpha }^F$ corresponds to a plane graph that dominates $\Gamma $. The nonparallel condition is equivalent to the condition that the graph is simple.

We now apply the general discussion carried out above to prove Proposition 3.17.

3.4 Quasi-Fuchsian space and mating locus

The above discussion applies to the special case of quasi-Fuchsian space. This space is related to the mating locus for critically fixed anti-rational maps.

Let $\Gamma _d$ be the marked $d+1$-sided polygonal graph; that is, $\Gamma _d$ contains $d+1$ vertices $v_1,\ldots , v_{d+1}$ with edges $v_iv_{i+1}$ where indices are understood modulo $d+1$. We choose the most symmetric circle packing $\mathcal {P}_d$ realising $\Gamma _d$. More precisely, consider the ideal $(d+1)$-gon in $\mathbb {D}\cong \mathbb {H}^2$ with vertices at the $(d+1)$-st roots of unity. The edges of this ideal $(d+1)$-gon are arcs of $d+1$ circles and we label these circles as $\mathcal {P}_d := \{\mathbf {C}_1,\ldots , \mathbf {C}_{d+1}\}$, where we index them counterclockwise such that $\mathbf {C}_1$ passes through $1$ and $e^{2\pi i/(d+1)}$. Let $\mathbf {G}_d$ be the kissing reflection group associated to $\mathcal {P}_d$. Note that $\widetilde {\mathbf {G}}_d$ is a Fuchsian group. We remark that since any embedding of the polygonal graph $\Gamma _d$ into a graph $\Gamma $ abstractly dominating $\Gamma _d$ respects the plane structure, we have that $\textrm {AH}(\Gamma _d)=\overline {\mathcal {QC}(\Gamma _d)}$.

Recall that a graph $\Gamma $ is said to be Hamiltonian if there exists a cycle which passes through each vertex exactly once. Since a simple plane graph with $d+1$ vertices is Hamiltonian if and only if it dominates $\Gamma _d$, the following proposition follows immediately from Proposition 3.17.

Let $\Gamma $ be a marked Hamiltonian simple plane graph with a Hamiltonian cycle $C = (v_1,\ldots ,v_{d+1})$. Let $G:=\langle g_1,\ldots , g_{d+1}\rangle $ be a kissing reflection group with contact graph $\Gamma $. The Hamiltonian cycle divides the fundamental domain $\Pi \subseteq \widehat {\mathbb {C}}$ into two parts and we denote them as $\Pi ^+$ and $\Pi ^-$, where we assume that the Hamiltonian cycle is positively oriented on the boundary of $\Pi ^+$ and negatively oriented on the boundary of $\Pi ^-$ (see Figure 3.4). We denote

$$ \begin{align*}\Omega^\pm := \bigcup_{g\in G} g\cdot \Pi^\pm. \end{align*} $$

Since each $\Omega ^\pm $ is G-invariant, we have $\Lambda (G) = \partial \Omega ^+ = \partial \Omega ^-$.

As in Subsection 3.1, set

$$ \begin{align*}\Pi^{1,\pm} = \bigcup_{i=1}^{d+1} g_i \cdot \Pi^\pm,\ \textrm{and}\quad \Pi^{j+1,\pm} = \bigcup_{i=1}^{d+1} g_i \cdot \left(\Pi^{j,\pm} \setminus \overline{D}_i\right).\end{align*} $$

For consistency, we also set $\Pi ^{0, \pm } = \Pi ^\pm $. Then the arguments of the proof of Lemma 3.2 show that

$$ \begin{align*}\Pi^{l,\pm} = \bigcup_{\vert g\vert = l} g\cdot \Pi^\pm. \end{align*} $$

Proof. We shall prove by induction that the closures $\overline {\bigcup _{i=0}^n \Pi ^{i,+}}$ and $\overline {\bigcup _{i=0}^n \Pi ^{i,+}\setminus \overline {D}_j}$ are connected for all n and j.

Indeed, the base case is true as $\overline {\Pi ^+}$ and $\overline {\Pi ^+ \setminus \overline {D}_j}$ are connected for all j. Assume that $\overline {\bigcup _{i=0}^n \Pi ^{i,+}}$ and $\overline {\bigcup _{i=0}^n \Pi ^{i,+}\setminus \overline {D}_j}$ are connected for all j. Note that

$$ \begin{align*}\overline{\bigcup_{i=0}^{n+1} \Pi^{i,+}} = \overline{\Pi^{+} \cup \bigcup_{j=1}^{d+1} g_j \cdot \left(\bigcup_{i=0}^n \Pi^{i,+} \setminus \overline{D}_j\right)}. \end{align*} $$

By the induction hypothesis, $\overline {g_j \cdot (\bigcup _{i=0}^n \Pi ^{i,+} \setminus \overline {D}_j)}$ is connected. Since each

$$ \begin{align*}\overline{g_j \cdot \left(\bigcup_{i=0}^n \Pi^{i,+} \setminus \overline{D}_j\right)} \end{align*} $$

intersects $\overline {\Pi ^{+}}$ along an arc of $C_j = \partial D_j$, we have that $\overline {\bigcup _{i=0}^{n+1} \Pi ^{i,+}}$ is also connected.

Similarly,

$$ \begin{align*}\overline{\bigcup_{i=0}^{n+1} \Pi^{i,+}\setminus \overline{D}_j} = \overline{\Pi^+ \cup \bigcup_{k\neq j} g_k \cdot \left(\bigcup_{i=0}^n \Pi^{i,+} \setminus \overline{D}_k\right)} \end{align*} $$

is connected for any j.

Connectedness of $\overline {\Omega ^-}$ is proved in the same way.

Matings of function kissing reflection groups

We say that a kissing reflection group G with connected limit set is a function kissing reflection group if there is a component $\Omega _0$ of $\Omega (G)$ invariant under G. This terminology was traditionally used in complex analysis as one can construct Poincaré series, differentials and functions on it.

We say that a simple plane graph $\Gamma $ with n vertices is outerplanar if it has a face with all n vertices on its boundary. We also call this face the outer face. Note that a $2$-connected outerplanar graph is Hamiltonian with a unique Hamiltonian cycle. The following proposition characterises function kissing reflection groups in terms of their contact graphs.

For a function kissing reflection group G, we set $\Omega _b := \Omega (G)\setminus \Omega _0$, where $\Omega _0$ is G-invariant. Similarly, we shall use the notation $\Pi _b := \Pi \cap \Omega _b$ (see Figure 3.6) and $\Pi ^i_b := \Pi ^i \cap \Omega _b$, for $i\geq 0$ (see Lemma 3.2). We call the closure $\overline {\Omega _b} = \widehat {\mathbb {C}} \setminus \Omega _0$ the filled limit set for the function kissing reflection group G and denote it by $\mathcal {K}(G)$.

Figure 3.6 The limit set of a function kissing reflection group H with an outerplanar contact graph. The grey region represents the part $\Pi _b$ of the fundamental domain $\Pi (H)$ corresponding to the non-outer faces of $\Gamma (H)$. The kissing reflection group G shown in Figure 3.4 can be constructed by pinching a simple closed curve for H.

Let $G^\pm $ be two function kissing reflection groups with the same number of vertices in their contact graphs. We say that a kissing reflection group G is a geometric mating of $G^+$ and $G^-$ if we have

• • a decomposition $\Omega (G) = \Omega ^+(G) \sqcup \Omega ^-(G)$ with $\Lambda (G) = \partial \Omega ^+(G) = \partial \Omega ^-(G)$;

• • weakly type-preserving isomorphisms $\phi ^\pm : G^\pm \longrightarrow G$;

• • continuous surjections $\psi ^{\pm }: \mathcal {K}(G^\pm ) \longrightarrow \overline {\Omega ^\pm (G)}$ which are conformal between the interior $\mathring {\mathcal {K}}(G^\pm )$ and $\Omega ^\pm (G)$ such that for any $g\in G^\pm $, $\psi ^\pm \circ g|_{\mathcal {K}(G^\pm )} = \phi ^\pm (g) \circ \psi ^\pm $.

Note that by the semi-conjugacy relation, the sets $\Omega ^\pm (G)$ are $G-$invariant. We shall now complete the proof of the group part of Theorem 1.2.

Proof. If G is a geometric mating of $G^\pm $, then we have a decomposition $\Omega (G) = \Omega ^+(G) \sqcup \Omega ^-(G)$ with $\Lambda (G) = \partial \Omega ^+(G) = \partial \Omega ^-(G)$. This gives a decomposition of $\Pi (G) = \Pi ^+ \sqcup \Pi ^-$ and thus a decomposition of the contact graph $\Gamma (G) = \Gamma ^+ \cup \Gamma ^-$. More precisely, the vertex sets of $\Gamma ^\pm $ coincide with that of $\Gamma (G)$ and there is an edge in $\Gamma ^\pm $ connecting two vertices if and only if the point of intersection of the corresponding two circles lies in $\partial \Pi ^\pm $. Since the action of $G^\pm $ on $\mathcal {K}(G^\pm )$ is semi-conjugate to G on $\overline {\Omega ^\pm (G)}$, it follows that $\Gamma ^\pm $ is isomorphic to $\Gamma (G^\pm )$ as plane graphs. Thus, the intersection of $\Gamma ^+$ and $\Gamma ^-$ (which is the common boundary of the outer faces of $\Gamma ^+$ and $\Gamma ^-$) gives a Hamiltonian cycle for $\Gamma (G)$.

Conversely, if $\Gamma (G)$ is Hamiltonian, then a Hamiltonian cycle yields a decomposition $\Gamma (G) = \Gamma ^+ \cup \Gamma ^-$, where $\Gamma ^+$ and $\Gamma ^-$ are outerplanar graphs with vertex sets equal to that of $\Gamma (G)$. We construct function kissing reflection groups $G^\pm $ with contact graphs $\Gamma (G^\pm ) = \Gamma ^\pm $. Note that we have a natural embedding of $\Gamma (G^\pm )$ into $\Gamma (G)$ (as plane graphs), which gives an identification of vertices and non-outer faces. Thus, we have weakly type-preserving isomorphisms $\phi ^\pm : G^\pm \longrightarrow G$ coming from the identification of vertices and Proposition 3.17. By quasiconformal deformation, we may assume that for each non-outer face $F\in \Gamma (G^\pm )$, the associated conformal boundary component $R_F(G^\pm )\subseteq \partial \mathcal {M}(G^\pm )$ is conformally equivalent to the corresponding conformal boundary component $R_F(G)\subseteq \partial \mathcal {M}(G)$.

The existence of the desired continuous map $\psi ^+$ can be derived from a more general statement on the existence of Cannon–Thurston maps (see [Reference Mj and Series32, Theorem 4.2]). For completeness and making the proof self-contained, we give an explicit construction of $\psi ^+: \mathcal {K}(G^+) \longrightarrow \overline {\Omega ^+(G)}$. This construction is done in levels: we start with a homeomorphism

$$ \begin{align*}\psi^+_0: \overline{\Pi^{0}_b(G^+)} \longrightarrow \overline{\Pi^{0, +}}.\end{align*} $$

This can be chosen to be conformal on the interior as the associated conformal boundaries are assumed to be conformally equivalent. Assuming that

$$ \begin{align*}\psi^+_i: \overline{\bigcup_{j=0}^i \Pi^{j}_b(G^+)} \longrightarrow \overline{\bigcup_{j=0}^i \Pi^{j,+}}\end{align*} $$

is constructed, we extend $\psi ^+_i$ to $\psi ^+_{i+1}$ by setting

$$ \begin{align*}\psi^+_{i+1} (z) := \phi^+(g_k)\circ \psi^+_i \circ g_k(z) \end{align*} $$

if $z\in \Pi ^{i+1}_b(G^+) \cap D_k$. It is easy to check by induction that $\psi ^+_{i+1}$ is a homeomorphism which is conformal on the interior (see Figure 3.4 and Figure 3.6).

Let $P^{0}(G^+) := \overline {\Pi ^{0}_b(G^+)}\setminus \Pi ^{0}_b(G^+)$; then $P^{0}(G^+)$ consists of cusps where various components of $\Pi ^{0}_b(G^+)$ touch. Let $P^{\infty }(G^+):= \bigcup _{g\in G^+} g\cdot P^{0}(G^+)$. Then $P^{\infty }(G^+)$ is dense in $\Lambda (G^+)$. We define $P^{\infty , +}$ similarly (where, $P^{0,+}:=\overline {\Pi ^{0,+}}\setminus \Pi ^{0,+}$) and note that $P^{\infty , +}$ is dense in $\partial \Omega ^+(G) = \Lambda (G)$.

Note that $\psi ^+_i = \psi ^+_j$ on $P^{i}(G^+)$ for all $j \geq i$. Thus, we have a well-defined limit $\psi ^+ : P^{\infty }(G^+) \longrightarrow P^{\infty , +}$. We claim that $\psi ^+$ is uniformly continuous on $P^{\infty }(G^+)$. For an arbitrary $\epsilon>0$, there exists N such that all disks in $\mathcal {D}^N(G)$ have spherical diameter $< \epsilon $ by Lemma 3.3. Choose $\delta $ so that any two nonadjacent disks in $\mathcal {D}^N(G^+)$ are separated in spherical metric by $\delta $. Then if $x,y \in P^{\infty }(G^+)$ with $d(x,y) < \delta $, they must lie in two adjacent disks of $\mathcal {D}^N(G^+)$. Thus, $\psi ^+_i(x)$ and $\psi ^+_i(y)$ (whenever defined) lie in two adjacent disks of $\mathcal {D}^N(G)$ for all $i\geq N$. Hence, $d(\psi ^+_i(x), \psi ^+_i(y)) < 2\epsilon $ for all $i \geq N$. This shows that $\psi ^+$ is uniformly continuous on $P^{\infty }(G^+)$.

Thus, we have a continuous extension $\psi ^+: \mathcal {K}(G^+) \longrightarrow \overline {\Omega ^+(G)}$. It is easy to check that $\psi ^+$ is surjective, conformal on the interior and equivariant with respect to the actions of $G^+$ and G.

The same proof gives the construction for $\psi ^-$. This shows that G is a geometric mating of $G^+$ and $G^-$.

Note that from the proof of Proposition 3.21, we see that each (marked) Hamiltonian cycle H gives a decomposition of the graph $\Gamma = \Gamma ^+ \cup \Gamma ^-$ and hence an unmating of a kissing reflection group. From the pinching perspective as in the proof of Proposition 3.17, this (marked) Hamiltonian cycle also gives a pair of nonparallel multicurves. This pair of multicurves can be constructed explicitly from the decomposition of $\Gamma $. Indeed, each edge in $\Gamma ^\pm - H$ gives a pair of nonadjacent vertices in H. The corresponding $\sigma $-invariant curve comes from these nonadjacent vertices in H. If a marking is not specified on H, this pair of multicurves is defined only up to simultaneous change of coordinates on the two conformal boundaries.

3.5 Nielsen maps for kissing reflection groups

Let $\Gamma $ be a simple plane graph and $G_\Gamma $ be a kissing reflection group with contact graph $\Gamma $. Recall that $\overline {\mathcal {D}} = \bigcup _{j=1}^n \overline {D_j}$ is defined as the union of the closure of the disks for the associated circle packing. We define the Nielsen map $\mathcal {N}_\Gamma : \overline {\mathcal {D}} \longrightarrow \widehat {\mathbb {C}}$ by

$$ \begin{align*}\mathcal{N}_\Gamma(z) = g_j(z) \text{ if } z\in \overline{D_j}. \end{align*} $$

Let us now assume that $\Gamma $ is $2$-connected; then the limit set $\Lambda (G_\Gamma )$ is connected. Thus, each component of $\Omega (G_\Gamma )$ is a topological disk. Let F be a face with $d+1$ sides of $\Gamma $ and let $\Omega _F$ be the component of $\Omega (G)$ containing $\Pi _F$. By a quasiconformal deformation, we can assume that the restriction of $G_\Gamma $ on $\Omega _F$ is conformally conjugate to the regular ideal $d+1$-gon reflection group on $\mathbb {D} \cong \mathbb {H}^2$.

For the regular ideal $d+1$-gon reflection group $\mathbf {G}_d$ (whose associated contact graph is $\Gamma _d$), the $d+1$ disks yield a Markov partition for the action of the Nielsen map $\mathcal {N}_d\equiv \mathcal {N}_{\Gamma _d}$ on the limit set $\Lambda (\mathbf {G}_d)=\mathbb {S}^1$. The diameters of the preimages of these disks under $\mathcal {N}_d$ shrink to $0$ uniformly by Lemma 3.3 and hence the Nielsen map $\mathcal {N}_d$ is topologically conjugate to $z\mapsto \overline {z}^d$ on $\mathbb {S}^1$ (cf. [Reference Lodge, Lyubich, Merenkov and Mukherjee24, §4]).

Therefore, the Nielsen map $\mathcal {N}_\Gamma $ is topologically conjugate to $\overline {z}^d$ on the ideal boundary of $\Omega _F$. This allows us to replace the dynamics of $\mathcal {N}_\Gamma $ on $\Omega _F$ by $\overline {z}^d$ for every face of $\Gamma $ and obtain a globally defined orientation-reversing branched covering $\mathcal {G}_\Gamma : \widehat {\mathbb {C}}\longrightarrow \widehat {\mathbb {C}}$.

More precisely, let $\phi :\mathbb {D} \longrightarrow \Omega _F$ be a conformal conjugacy between $\mathcal {N}_d$ and $\mathcal {N}_\Gamma $, which extends to a topological conjugacy from the ideal boundary $\mathbb {S}^1 = I(\mathbb {D}) = \partial \mathbb {D}$ onto $I(\Omega _F)$. Let $\psi : \overline {\mathbb {D}} \longrightarrow \overline {\mathbb {D}}$ be an arbitrary homeomorphic extension of the topological conjugacy between $\overline {z}^d\vert _{\mathbb {S}^1}$ and $\mathcal {N}_d\vert _{\mathbb {S}^1}$ fixing $1$. We define

(3.2)$$ \begin{align} \mathcal{G}_\Gamma :=\left\{\begin{array}{ll} \mathcal{N}_\Gamma & \mbox{on}\ \overline{\mathcal{D}}\setminus \bigcup_{F} \Omega_F, \\ (\phi\circ\psi) \circ m_{-d} \circ (\phi\circ\psi)^{-1} & \mbox{on}\ \Omega_F, \end{array}\right. \end{align} $$

where $m_{-d}(z) = \overline {z}^d$.

The map $\overline {z}^d$ has $d+1$ invariant rays in $\mathbb {D}$ connecting $0$ with $e^{2\pi i \cdot \frac {j}{d+1}}$ by radial line segments. We shall refer to these rays as internal rays. For each $\Omega _F$, we let $\mathscr {T}_F$ be the image of the union of these $d+1$ internal rays (under $\phi \circ \psi $), and we call the image of $0$ the centre of $\Omega _F$. This graph $\mathscr {T}_F$ is a deformation retract of $\Pi _F$ fixing the ideal points. Thus, each ray of $\Omega _F$ lands exactly at a cusp where two circles of the circle packing touch. We define

(3.3)$$ \begin{align} \mathscr{T}(\mathcal{G}_\Gamma) := \bigcup_F \overline{\mathscr{T}_F} \end{align} $$

and endow it with a simplicial structure such that the centres of the faces are vertices. Then $\mathscr {T}(\mathcal {G}_\Gamma )$ is the planar dual to the plane graph $\Gamma $. We will now see that $\mathcal {G}_\Gamma $ acts as a ‘reflection’ on each face of $\mathscr {T}(\mathcal {G}_\Gamma )$.

We shall see in the next section that this branched covering $\mathcal {G}_\Gamma $ is topologically conjugate to a critically fixed anti-rational map R and this dual graph $\mathscr {T}(\mathcal {G}_\Gamma )$ is the Tischler graph associated with R.

4.1 Tischler graph of critically fixed anti-rational maps

Let R be a critically fixed anti-rational map of degree d. Let $c_1,\cdots , c_k$ be the distinct critical points of R and the local degree of R at $c_i$ be $m_i$ ($i=1,\cdots , k$). Since R has $(2d-2)$ critical points counting multiplicity, we have that

$$ \begin{align*}\displaystyle\sum_{i=1}^k (m_i-1)=2d-2\ \implies\ \displaystyle\sum_{i=1}^k m_i=2d+k-2. \end{align*} $$

Suppose that $U_i$ is the invariant Fatou component containing $c_i$ ($i=1,\cdots , k$). Then $U_i$ is a simply connected domain such that $R\vert _{U_i}$ is conformally conjugate to $\overline {z}^{m_i}\vert _{\mathbb {D}}$ [Reference Milnor31, Theorem 9.3]. This defines internal rays in $U_i$ and R maps the internal ray at angle $\theta \in \mathbb {R}/\mathbb {Z}$ to the one at angle $-m_i\theta \in \mathbb {R}/\mathbb {Z}$. It follows that there are $(m_i+1)$ fixed internal rays in $U_i$. A straightforward adaptation of [Reference Milnor31, Theorem 18.10] now implies that all of these fixed internal rays land at repelling fixed points on $\partial U_i$.

We define the Tischler graph $\mathscr {T}$ of R as the union of the closures of the fixed internal rays of R.

Proof. As R is a local orientation-reversing diffeomorphism in a neighbourhood of the landing point of each internal ray, it follows that at most two distinct fixed internal rays may land at a common point.

Note that the total number of fixed internal rays of R is

$$ \begin{align*}\displaystyle\sum_{i=1}^k (m_i+1)=2(d+k-1), \end{align*} $$

while there are only $(d+k-1)$ landing points available for these rays by Lemma 4.1. Since no more than two fixed internal rays can land at a common fixed point, the result follows.

In our setting, it is more natural to put a simplicial structure on $\mathscr {T}$ so that the vertices correspond to the critical points of R. We will refer to the repelling fixed points on $\mathscr {T}$ as the midpoints of the edges. To distinguish an edge of $\mathscr {T}$ from an arc connecting a vertex and a midpoint, we will call the latter an internal ray of $\mathscr {T}$.

The proof of Lemma 4.2 implies the following (see [Reference Hlushchanka11, Corollary 6] for the same statement in the holomorphic setting).

We are now ready to establish the key properties of the Tischler graph of a critically fixed anti-rational map that will be used in the combinatorial classification of such maps.

Proof. Let F be a face of the Tischler graph $\mathscr {T}$. Then a component of the ideal boundary $I(F)$ consists of a sequence of edges $e_1,\ldots , e_m$ of $\mathscr {T}$, oriented counterclockwise viewed from F. Note that a priori, $e_i$ may be equal to $e_j$ in $\mathscr {T}$ for different i and j.

Consider the graph T whose vertices are components of $\widehat {\mathbb {C}}\setminus \overline {F}$ and two vertices $U, V$ are connected by an edge if there is a path in $\widehat {\mathbb {C}}\setminus F$ connecting $U,V$ and not passing through other components. Then T is a finite union of trees. Since each vertex (of $\mathscr {T}$) has valence at least $3$, we have that $e_i\neq e_{i+1}$. This also implies that two adjacent components $U,V$ either share a common boundary vertex or there exists an edge of $\mathscr {T}$ connecting them. Let U be a component of $\widehat {\mathbb {C}}\setminus \overline {F}$. If $e_i, e_j$ are a pair of adjacent edges of $\partial U$ but $e_i, e_j$ are not adjacent on the ideal boundary $I(F)$, then there exists a component V of $\widehat {\mathbb {C}}\setminus \overline {F}$ ‘attached’ to U through the vertex $v = e_i \cap e_j$ (see Figure 4.1). Therefore, the number of pairs of adjacent edges of $\partial U$ that are not adjacent on the ideal boundary $I(F)$ equals to the valence of U in T. Choose U to be an end point of T and let $S = \partial U$.

Figure 4.1 An a priori possible schematic of a component of the boundary of F.

Since R is orientation-reversing and each edge of $\mathscr {T}$ is invariant under the map, R reflects the two sides near each open edge of S. Thus, there is a connected component V of $R^{-1}(U)$ with $\partial V \cap \partial U \neq \emptyset $ and $V \cap F \neq \emptyset $. It is easy to see from the local dynamics of $\overline {z}^{m_i}$ at the origin that near a critical point, R sends the region bounded by two adjacent edges of $\mathscr {T}$ to its complement. Therefore, by our choice of U, we have $\partial U \subseteq \partial V$. Since $\partial F$ is R-invariant, $V \cap \partial F = \emptyset $. Thus, $V \subseteq F$ and V contains no critical points of R. As U is a disk, the Riemann–Hurwitz formula now implies that V is a disk and R is a homeomorphism from $\partial V$ to $\partial U$. Therefore, $\partial U = \partial V$ and $V = F$; so F is a Jordan domain.

In particular, we have the following.

In fact, the Tischler graph gives a topological model for the map R.

The following result, where we translate the above properties of the Tischler graph to a simple graph-theoretic property of its planar dual, plays a crucial role in the combinatorial classification of critically fixed anti-rational maps.

4.2 Constructing critically fixed anti-rational maps from graphs

Let $\Gamma $ be a $2$-connected simple plane graph and $G_\Gamma $ be an associated kissing reflection group. In Subsection 3.5, we constructed a topological branched covering $\mathcal {G}_\Gamma $ from the Nielsen map of $G_\Gamma $. In the remainder of this section, we will use this branched covering $\mathcal {G}_\Gamma $ to promote Lemma 4.9 to a characterisation of Tischler graphs of critically fixed anti-rational maps.

We will first introduce some terminology. A postcritically finite branched covering (possibly orientation-reversing) of a topological $2$-sphere $\mathbb {S}^2$ is called a Thurston map. We denote the postcritical set of a Thurston map f by $P(f)$. Two Thurston maps f and g are equivalent if there exist two orientation-preserving homeomorphisms $h_0,h_1:(\mathbb {S}^2,P(f))\to (\mathbb {S}^2,P(g))$ so that $h_0\circ f = g\circ h_1$, where $h_0$ and $h_1$ are isotopic relative to $P(f)$.

A set of pairwise disjoint, nonisotopic, essential, simple closed curves $\Sigma $ on $\mathbb {S}^2\setminus P(f)$ is called a curve system. A curve system $\Sigma $ is called f-stable if for every curve $\sigma \in \Sigma $, all of the essential components of $f^{-1}(\sigma )$ are homotopic rel $P(f)$ to curves in $\Sigma $. We associate to an f-stable curve system $\Sigma $ the Thurston linear transformation

$$ \begin{align*}f_\Sigma: \mathbb{R}^\Sigma \longrightarrow \mathbb{R}^{\Sigma} \end{align*} $$

defined as

$$ \begin{align*}f_\Sigma(\sigma) = \sum_{\sigma' \subseteq f^{-1}(\sigma)} \frac{1}{\deg(f: \sigma' \rightarrow \sigma)}[\sigma']_\Sigma, \end{align*} $$

where $\sigma \in \Sigma $ and $[\sigma ']_\Sigma $ denotes the element of $\Sigma $ isotopic to $\sigma '$, if it exists. The curve system is called irrreducible if $f_\Sigma $ is irreducible as a linear transformation. It is said to be a Thurston obstruction if the spectral radius $\lambda (f_\Sigma ) \geq 1$.

Similarly, an arc $\lambda $ in $\mathbb {S}^2\setminus P(f)$ is an embedding of $(0,1)$ in $\mathbb {S}^2\setminus P(f)$ with endpoints in $P(f)$. It is said to be essential if it is not contractible in $\mathbb {S}^2$ fixing the two endpoints. A set of pairwise nonisotopic essential arcs $\Lambda $ is called an arc system. The Thurston linear transformation $f_\Lambda $ is defined in a similar way, and we say that it is irreducible if $f_\Lambda $ is irreducible as a linear transformation.

The next proposition allows us to directly apply Thurston’s topological characterisation theorem for rational maps (see [Reference Douady and Hubbard6, Theorem 1]) to the study of orientation-reversing Thurston maps (see [Reference Geyer9, Theorem 3.9] for an intrinsic topological characterisation theorem for anti-rational maps). It is proved by considering the second iterate of Thurston’s pullback map on the Teichmüller space of $\mathbb {S}^2\setminus P(f)$. The reader is referred to [Reference Douady and Hubbard6] for the definition of hyperbolic orbifold, but this is the typical case as any map with more than four postcritical points has hyperbolic orbifold.

For a curve system $\Sigma $ (respectively an arc system $\Lambda $), we set $\widetilde {\Sigma }$ (respectively $\widetilde {\Lambda }$) as the union of those components of $f^{-1}(\Sigma )$ (respectively $f^{-1}(\Lambda )$) which are isotopic relative to $P(f)$ to elements of $\Sigma $ (respectively $\Lambda $). We will use $\Sigma \cdot \Lambda $ to denote the minimal intersection number between them. We will be using the following theorem excerpted and paraphrased from [Reference Pilgrim and Tan38, Theorem 3.2].

With these preparations, we are now ready to show that $\mathcal {G}_\Gamma $ is equivalent to an anti-rational map. The proof is similar to [Reference Lodge, Lyubich, Merenkov and Mukherjee24, Proposition 6.2].

Proof. It is easy to verify that $\mathcal {G}_\Gamma \circ \mathcal {G}_\Gamma $ has hyperbolic orbifold, except when $\Gamma =\Gamma _d$, in which case $\mathcal {G}_\Gamma $ has only two fixed critical points, each of which is fully branched and $\mathcal {G}_\Gamma $ is equivalent to $\overline {z}^d$. Thus, by Proposition 4.11, we will prove the lemma by showing that there is no Thurston obstruction for $\mathcal {G}_\Gamma \circ \mathcal {G}_\Gamma $.

We will assume that there is a Thurston obstruction $\Sigma $ and arrive at a contradiction. After passing to a subset, we may assume that $\Sigma $ is irreducible. Isotoping the curve system $\Sigma $, we may assume that $\Sigma $ intersects the graph $\mathscr {T}(\mathcal {G}_\Gamma )$ minimally (see Subsection 3.5 for the definition of $\mathscr {T}(\mathcal {G}_\Gamma )$). Let $\lambda $ be an edge in $\mathscr {T}(\mathcal {G}_\Gamma )$; then $\Lambda = \{\lambda \}$ is an irreducible arc system. Let $\widetilde {\Sigma }$ be the union of those components of $\mathcal {G}_\Gamma ^{-2}(\Sigma )$ which are isotopic to elements of $\Sigma $.

We claim that $\widetilde {\Sigma }$ does not intersect $\mathcal {G}_\Gamma ^{-2}(\lambda ) \setminus \lambda $. Indeed, applying Theorem 4.12, we are led to the following two cases. In the first case, we have $\Sigma \cdot \lambda = 0$; hence, $\mathcal {G}_\Gamma ^{-2}(\Sigma )$ cannot intersect $\mathcal {G}_\Gamma ^{-2}(\lambda )$ and the claim follows. In the second case, since $\mathcal {G}_\Gamma (\lambda ) = \lambda $, we conclude that $\lambda $ is the unique component of $\mathcal {G}_\Gamma ^{-2}(\lambda )$ isotopic to $\lambda $. Thus, the only component of $\mathcal {G}_\Gamma ^{-2}(\lambda )$ intersecting $\widetilde {\Sigma }$ is $\lambda $, so the claim follows.

Applying this argument on all the edges of $\mathscr {T}(\mathcal {G}_\Gamma )$, we conclude that $\widetilde {\Sigma }$ does not intersect $\mathcal {G}_\Gamma ^{-2}(\mathscr {T}(\mathcal {G}_\Gamma )) \setminus \mathscr {T}(\mathcal {G}_\Gamma )$.

Let F be a face of $\mathscr {T}(\mathcal {G}_\Gamma )$ with vertices of $\partial F$ denoted by $v_1, \ldots , v_k$ counterclockwise. Denote $e_i \subseteq \partial F$ be the open edge connecting $v_i, v_{i+1}$. We claim that the graph obtained by removing the boundary edges of F

$$ \begin{align*}\mathscr{T}_{F}:=\mathscr{T}(\mathcal{G}_\Gamma) \setminus \bigcup_{i=1}^k e_i \end{align*} $$

is still connected. Since $\mathscr {T}(\mathcal {G}_\Gamma )$ is connected, any vertex v is connected to the set $\{v_1,\ldots , v_k\}$ in $\mathscr {T}_{F}$. Thus, it suffices to show that $v_i$ and $v_{i+1}$ are connected in $\mathscr {T}_{F}$ for all i. As the planar dual $\Gamma $ of $\mathscr {T}(\mathcal {G}_\Gamma )$ is $2$-connected and has no self-loop, each face of $\mathscr {T}(\mathcal {G}_\Gamma )$ is a Jordan domain. Hence, there exists a face $F_i\neq F$ of $\mathscr {T}(\mathcal {G}_\Gamma )$ sharing $e_i$ on the boundary with F. Moreover, since $\Gamma $ has no multi-edge, the boundaries of any two faces of $\mathscr {T}(\mathcal {G}_\Gamma )$ intersect at most at one edge. Thus, $\partial F_i \cap \partial F = e_i \cup \{v_i, v_{i+1}\}$. Hence, $v_i$ and $v_{i+1}$ are connected by a path in $\partial F_i \setminus e_i \subseteq \mathscr {T}_{F}$, and this proves the claim.

Therefore, by Lemma 3.22, we deduce that

$$ \begin{align*}(\mathcal{G}_\Gamma^{-1} (\mathscr{T}(\mathcal{G}_\Gamma)) \setminus \mathscr{T}(\mathcal{G}_\Gamma)) \cup V(\mathscr{T}(\mathcal{G}_\Gamma)) \end{align*} $$

is connected. Thus, $(\mathcal {G}_\Gamma ^{-2}(\mathscr {T}(\mathcal {G}_\Gamma )) \setminus \mathscr {T}(\mathcal {G}_\Gamma )) \cup V(\mathscr {T}(\mathcal {G}_\Gamma ))$ is a connected graph containing the postcritical set $P(\mathcal {G}_\Gamma \circ \mathcal {G}_\Gamma ) = V(\mathscr {T}(\mathcal {G}_\Gamma ))$. This forces $\Sigma $ to be empty, which is a contradiction.

Therefore, $\mathcal {G}_\Gamma $ is equivalent to an anti-rational map $\mathcal {R}_\Gamma $. Since $\mathcal {G}_\Gamma $ is critically fixed, so is $\mathcal {R}_\Gamma $.

Let $h_0, h_1:(\mathbb {S}^2,P(\mathcal {G}_\Gamma ))\to (\mathbb {S}^2,P(\mathcal {R}_\Gamma ))$ be two orientation-preserving homeomorphisms so that $h_0\circ \mathcal {G}_\Gamma = \mathcal {R}_\Gamma \circ h_1$ where $h_0$ and $h_1$ are isotopic relative to $P(\mathcal {G}_\Gamma ) = V(\mathscr {T}(\mathcal {G}_\Gamma ))$.

We now use a standard pullback argument to prove the following.

Proof. Let $\alpha $ be an edge of $\mathscr {T}(\mathcal {G}_\Gamma )$; then $\mathcal {G}_\Gamma (\alpha ) = \alpha $. Consider the sequence of homeomorphisms $h_i$ with

$$ \begin{align*}h_{i-1} \circ \mathcal{G}_\Gamma = \mathcal{R}_\Gamma \circ h_i. \end{align*} $$

Each $h_i$ is normalised so that it carries $P(\mathcal {G}_\Gamma )$ to $P(\mathcal {R}_\Gamma )$. Note by induction, $\beta _i = h_i(\alpha )$ is isotopic to $\beta _0$ relative to the endpoints. Applying isotopy, we may assume that there is a decomposition $\beta _0 = \beta _0^- + \gamma _0 + \beta _0^+$, where $\beta _0^\pm $ are two internal rays and $\gamma _0$ does not intersect any invariant Fatou component. Then every $\beta _i = \beta _i^- + \gamma _i + \beta _i^+$ has such a decomposition, too.

Note that $\mathcal {R}_\Gamma (\gamma _{i+1}) = \gamma _{i}$ and $\mathcal {R}_\Gamma :\gamma _{i+1}\to \gamma _{i}$ is injective for each i. Since $\mathcal {R}_\Gamma $ is hyperbolic and the curves $\gamma _i$ are uniformly bounded away from the postcritical point set of $\mathcal {R}_\Gamma $, we conclude that the sequence of curves $\{\gamma _i\}$ shrinks to a point. Thus, if $\beta $ is a limit of $\beta _i$ in the Hausdorff topology, then $\beta $ is a union of two internal rays and it is isotopic to $\beta _0$. Therefore, $h_0(\alpha )$ is isotopic to an edge of $\mathscr {T}(\mathcal {R}_\Gamma )$.

Since this is true for every edge of $\mathscr {T}(\mathcal {G}_\Gamma )$ and $h_0$ is an orientation-preserving homeomorphism, by counting the valence at every critical point, we conclude that $h_0(\mathscr {T}(\mathcal {G}_\Gamma ))$ is isotopic to $\mathscr {T}(\mathcal {R}_\Gamma )$.